arXiv subjects
Publications and source records attributed to Rick Miranda.
Let R be any subring of the reals. We present a generalization of linear systems on graphs where divisors are R-valued functions on the set of vertices and graph edges are permitted to have nonegative weights in R. Using this generalization, we provide an independent proof of a Riemann-Roch formula, which implies the Riemann-Roch formula of Baker and Norine.
In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.
We prove that, for the general curve of genus g, the 2nd Gaussian map is injective if g <= 17 and surjective if g >= 18. The proof relies on the study of the limit of the 2nd Gaussian map when the general curve of genus g degenerates to a general stable binary curve, i.e. the union of two rational curves meeting at g+1 points.
In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.
In this paper we prove that for all pairs $(d,m)$ with $d/m \geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension.
This paper surveys and gives a uniform exposition of results contained in papers published by the team of authors. The subject is degenerations of surfaces, especially to unions of planes. More specifically, we deduce some properties of the smooth surface which is the general fibre of the degeneration from combinatorial features of the central fibre. In particular we show that there are strong constraints on the invariants of a smooth surface which degenerates to configurations of planes. Finally we consider several examples of embedded degenerations of smooth surfaces to unions of planes. Our interest in these problems has been raised by a series of interesting articles by Guido Zappa in 1950's.
In this paper we study smooth, non-special scrolls S of degree d, genus g, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques.
In this paper we study degenerations of scrolls to union of planes, a problem already considered by G. Zappa in 1940-50. We prove, using techniques different from the ones of Zappa, a degeneration result to union of planes with the mildest possible singularities, for linearly normal scrolls of genus $g$ and of degree $d$ larger than $2g+4$ in $\Pp^{d-2g+1}$. We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally we review Zappa's original approach.
In this article we construct a specific projective degeneration of K3 surfaces of degree 2g-2 in P^g to a union of 2g-2 planes, which meet in such a way that the combinatorics of the configuration of planes is a triangulation of the 2-sphere. Abstractly, such degenerations are said to be Type III degenerations of K3 surfaces. Although the birational geometry of such degenerations is fairly well understood, the study of projective degenerations is not nearly as completely developed. In this article we construct degenerations for which the general member is embedded by a multiple of the primitive line bundle class.
Remarks on the Obstructedness of Cones Over Curves of Low Genus. Reason for replacement: proof of the main lemma was significantly improved.
In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which $-K$ is very ample. An argument for the associativity in general is proposed, which also avoids resorting to the symplectic category. The enumerative predictions of Kontsevich and Manin concerning the degree of the rational curve locus in a linear system are recovered. The associativity of the quantum product for the cubic surface is shown to be essentially equivalent to the classical enumerative facts concerning lines: there are $27$ of them, each meeting $10$ others.
In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$. We also prove that the general such hyperplane section lies on a unique $K3$ surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.
Given a torsion section of a semistable elliptic surface, we prove equidistribution results for the components of singular fibers which are hit by the section, and for the root of unity (identifying the zero component with ${\Bbb C}$) which is hit by the section in case the section hits the zero component.
Let S be a torsion section of an elliptic surface with only I_n fibers. This article addresses the question: which components of singular fibers can S pass through? We give necessary criteria for the "component numbers", and show an equidistribution result for torsion sections of prime order.
Toric varieties are a special class of rational varieties defined by equations of the form {\it monomial = monomial}. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety $X$ contains a cover by affine open sets described in terms of arrangements (called fans) of convex bodies in $\Bbb R^r$. The coordinate rings of each of these affine open sets is a graded ring generated over the ground field by monomials. As a consequence, toric varieties provide a good context in which cohomology can be calculated. The purpose of this article is to describe the second étale cohomology group with coefficients in the sheaf of units of any toric variety $X$. This is the so-called cohomological Brauer group of $X$.